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A unified model for adhesive interfaces with damage, viscosity and friction
Gianpietro del Piero, Michel Raous
To cite this version:
Gianpietro del Piero, Michel Raous. A unified model for adhesive interfaces with damage, viscosity and
friction. European Journal of Mechanics - A/Solids, Elsevier, 2010, 29 (4), pp.496-507. �hal-00462116�
A unified model for adhesive interfaces with damage, viscosity, and friction
Gianpietro Del Piero
Dipartimento di Ingegneria, Università di Ferrara, via Saragat 1, 44100 Ferrara, Italy [email protected]
Michel Raous
Laboratoire de Mécanique et d’Acoustique, CNRS, 31 Chemin Joseph Aiguier 13402, Marseille cedex 20, France [email protected]
(Corresponding Author. Phone: 0033 491 16 40 53, Fax: 0033 491 16 44 81) Abstract
A general framework for models describingadhesive contact between rigid bodies is proposed. The intensity of adhesion is supposed to decrease under the action of prescribed tangential and normal relative displacements. The reduction is attributed to progressive damage, and comes with energy dissipation. Additional dissipation due to viscosity and friction is also taken into account. The response of the interface is described by a single state variable. It is determined by general laws expressing a mechanical version of the first two laws of thermodynamics, combined with a set of phenomenological assumptions.
Key words: Adhesive contact, cohesive interface.
1 – Introduction
During the last few decades, several models for cohesive interfaces have been developed. Many papers deal with cohesive zone models in fracture mechanics (Dugdale, 1960; Barenblatt, 1962; Needleman, 1987, 1990, 1992;
Tvergaard, 1990; Tvergaard and Hutchinson, 1992, 1996; Xu and Needleman, 1994; Costanzo and Walton, 1997; Needleman and Rosakis, 1999; Monerie, 2000; Péralès, 2005; Brinckmann and Siegmund, 2008). Other models are devoted to specific modes of interface decohesion in composite materials, such as delamination (Allix et al, 1995), peeling (Nguyen and Levy, 2009), or fiber debonding (Michel and Suquet, 1994; Raous et al, 1999; Monerie, 2000; Raous and Monerie, 2002). In Geophysics, the two main classes of interface models for studying fault nucleation, “rate and state” and “slip weakening” laws (Campillo and Ionescu, 1997; Rice and Ruina, 1983; Uenishi and Rice, 2003), can also be considered among the cohesive zone models. Finally, there is a number of models dealing in a more general way with adhesive interfaces (Frémond, 1987, 1988; Raous, 1999; Raous et al, 1999; Chaboche et al, 2001; Raous and Monerie, 2002; Talon and Curnier, 2003; Freddi and Frémond, 2006). This list is by no means exhaustive; other references can be found in the papers cited above.
In most theories, the loss of cohesion is described by a damage variable, called intensity of adhesion by Frémond (1987, 1988). This variable is defined on the interface, considered as a material boundary with a null thickness. Initially, adhesion was studied only under monotonic loading conditions, so that the problem of the reversibility of the material response was not taken into consideration. Only after the 1990’s, with the study of cyclic behavior, it became standard to consider damage as irreversible. An exception to this generally accepted trend is the model of adhesion with healing recently proposed by Raous et al (2006), in which adhesion is supposed to recover, partially or totally, when contact is restored after total separation.
In the contributions cited above, different assumptions have been made regarding the main characteristics of the interface behavior:
- for normal relative displacements, the non-interpenetration condition has been enforced either by unilateral conditions, or by compliance or penalization devices,
- for tangential relative displacements, either frictional or frictionless contact has been considered; in frictional models, the transition between the initial regime of full adhesion and the final purely frictional regime has been assumed to be either progressive or brutal,
- some models consider rate dependent effects such as viscosity, while other ones are rate independent, - the coupling between normal and tangential effects is often underestimated. This is mainly because in most
applications one of them is dominating.
The aim of the present paper is to provide a relatively general theoretical framework for adhesive contact. In our model the loss of adhesion is identified with irreversible damage, and the intensity of damage is assumed to depend on the coupled effect of normal and tangential deformation. Moreover, we assume unilateral conditions for normal displacements and Coulomb friction for tangential displacements. We also assume that the intensity of friction grows progressively with damage, and that the dissipation includes a part due to viscosity.
The proposed framework is based on the definition of a small number of fundamental objects:
• general laws, typically, energy conservation and dissipation principle, that is, mechanical versions of the first two laws of thermodynamics,
• a set of state variables, that is, an array of independent variables which fully determine the response to all possible deformation processes,
• a set of elastic potentials and dissipation potentials, which are functions of state in terms of which the general laws take specific forms,
• a set of constitutive assumptions.
The choice of the state variables, of the potentials, and of the constitutive equations is clearly influenced by the phenomenological features of the interface. It is at this level that specific microstructural schemes or failure mechanisms enter the play. We proceed to our construction systematically throughout Section 2. In Subsection 2.1 we start from the simplest case of a purely normal deformation with adhesion, in which adhesion is assumed to decrease progressively with increasing damage. Then we gradually enrich the initial scheme by introducing viscous dissipation in Subsection 2.2, and tangential deformation in Subsection 2.3. Finally, in Subsection 2.5 we discuss the case of coupled normal and tangential deformation in the presence of damage, viscosity, and friction.
The model is based on a single state variable which, together with the normal and tangential relative displacements at the interface, defines all elastic and dissipation potentials. The state variable measures the current intensity of damage, and is strictly related to the intensity of adhesion introduced by Frémond (1987, 1988). The choice of a single state variable reflects the idea that the intensity of damage is determined by the coupled effects of normal and tangential loading.
In the case of purely normal deformation, the basic experimental information that we assume to be known is the loading curve. This is the force-displacement relation in a monotonic deformation process starting from the virgin state. In the case of coupled normal and tangential deformation we show that, interestingly enough, the constitutive parameters are not determined by two separate loading curves, one for the normal and one for the tangential deformation, but by a single curve corresponding to an appropriate diagonal experiment.
Usually the power equation, which comes from time differentiation of the equation of energy conservation, is sufficient to determine the evolution of the state variable. However, there are situations in which the power equation is identically satisfied. Typically this is the case when dissipation due to damage is present, and viscous dissipation is neglected. In such situations, the evolution of the state variable is determined by the subsequent derivatives of the power equation. In the presence of physical constraints expressed by inequalities, for example, non-interpenetration, different responses at loading and unloading are obtained.
The efficiency of the theoretical scheme is tested on a specific process involving both normal and tangential deformations. This is done in Subsection 2.4 in the case of dissipation due to damage, and in Subsection 2.5 for dissipation due to damage, viscosity and friction. The response of the interface is determined qualitatively, without recourse to a numerical code, and the results are quite realistic. In the final Section 3 we show that, by an appropriate choice of the potentials and of the constitutive equations, the RCCM model developed in (Raous et al, 1999; Raous and Monerie, 2002), fits the proposed general scheme.
The RCCM model has been applied to more general situations, such as 2D and 3D deformable bodies instead of 2D rigid bodies (Raous et al, 1999; Raous and Monerie, 2002). Similar generalizations can be made for the model presented here. We do not discuss this possibility, since our effort is rather directed to the construction of a systematic model starting from a minimal number of general assumptions.
2. Construction of a general model for adhesive interfaces
Consider two rigid bodies separated by a plane material interface of negligible thickness. We take u and v to be the normal and tangential components, respectively, of the relative displacement of the surfaces of the two bodies. We also take σ and τ to be the normal and tangential components of the force transmitted across the interface. The restriction
u ≥ 0 (1)
is imposed to prevent interpenetration. We assume that a state of the interface is defined by the triplet (u, v, α), where α is a state variable measuring the current level of damage. Our purpose is to determine the evolution t α(t) of α in a deformation process t (u(t), v(t)) starting from a given initial state α(0) = α0. The response curves (σ, u) and (τ, v) then follow from the assumed constitutive equations.
We construct our model by steps, beginning from the case of purely normal loading, in which the tangential displacement and the tangential force are absent. We initially consider the simplest case in which the only source of dissipation is the damage due to the decrease in the adhesion intensity, and only later we will take into account the additional sources of dissipation provided by viscosity and friction.
2.1. Adhesion with purely normal damage
Consider an interface in which the damage is purely normal. In this case it is sufficient to prescribe a normal displacement u, determining a force σ normal to the contact surface. A deformation process then reduces to the function t u(t). We are interested in the following type of response:
(i) there is a function f which determines the force σ(t) = f(u(t)) at first loading, that is, in a monotonic non- decreasing process t u(t) from the virgin state (u(0) ,α(0))= (0, 0).
(ii) at unloading, σ decreases linearly down to the origin,
(iii) during the subsequent reloading process the unloading line is crossed backwards, and when the loading curve σ = f (u) is reached again, the reloading process continues along that curve,
(iv) there is a critical value ur of u such that, after this value has been reached, the force takes permanently the value σ = 0,
(v) when u = 0, a compressive force σ of arbitrary intensity can be transmitted across the surface.
We assume that the function f is defined for all u ≥ 0, positive in the open interval (0, ur) and null outside, and that it has the general shape shown in Fig. 1a. The laws assumed in the papers mentioned in the Introduction all have this general shape. Its restriction to (0, ur) is assumed to be star-shaped with respect to the origin. That is, every line from the origin has at most one intersection with this curve. Consequently, the function
g(u) := fu( (2) u) is strictly decreasing for u < ur and is identically zero for u ≥ ur. The function g measures the current stiffness of the interface, while the critical value ur corresponds to complete rupture of the interfacial bonds.
Figure 1. Adhesion with damage under normal loading. Response to a deformation process of loading- unloading starting from the origin, represented in the force-displacement plane (a), and in the state space (b).
There are only two types of pairs (σ, u) which are admissible for this type of response, in the sense that they can be reached in a deformation process t u(t) starting from the origin:
− those with 0 ≤ u < ur and 0 ≤σ ≤ f(u),
− those with u ≥ ur and σ= 0.
Though the behavior of the interface, as defined by items (i)-(v) above, is fully described in terms of the variables σ and u, in view of subsequent developments we find it convenient to introduce a state variable α, defined over the admissible pairs (σ, u) in the following way:
(a) (b)
u α ur u α
α
ΣΣΣΣ
ur
u α ur u σ
σ = f(u)
Ψ Ψ Ψ Ψ
∆∆∆∆
f(α) f(u)
. if
, 0
if / ) (
, 0 if
] , 0 [
r r
r r
u u u
u u u
g
u u
≥
=
<
<
=
=
∈
α σ
α α
(3) Notice the indeterminacy of α for u = 0, while, by the assumed monotonicity of g, the equation g(α) = σ / u defines α unambiguously for all u in (0, ur). Conversely, every pair (σ, α) with
σ < f(α) if α < ur , σ = 0 if α ≥ ur , defines one and only one pair (σ, u).
According to the definition given for α, the loading curve is made of points (σ, u) with u = α and σ = f(α).
For points (σ, u) not on the loading curve, α is the abscissa of the intersection of the loading curve with the line joining the point (σ, u) with the origin. The inequality
α ≥ u (4) holds at all points (σ, u) with u in (0, ur).
A pair (u, α) is a state of the interface. A state is admissible if the corresponding pair (σ, u) is admissible.
The set of all admissible states
Σ = { (u, α) | ur ≥ α ≥ u ≥ 0 } ∪ { (u, α) | α = ur , u≥ ur } , (5) is the state space.
In Fig.1a, the response to first loading up to α followed by unloading up to uis shown by a dotted line in the (σ, u) plane. From the figure it is clear that, before complete rupture is achieved, α represents the largest displacement reached in a deformation process up to the current instant. The state space is represented in Fig.1b, where the dotted line shows the trajectory in the state space corresponding to the trajectory in the force- displacement plane (σ, u). The dotted line follows the α = u line during the loading regime, and a horizontal line during the subsequent unloading regime.
In what follows the response described above in items (i)-(v) is described mathematically by specifying the evolution law t α(t) for the state variable. To do this we use the elastic potential (6), the dissipation potential (8), the general laws (10), (11), and the constitutive equation (19) below. This approach may look formal and of little use. In fact, its importance relies on fixing a systematic procedure, to be applied later to far more general situations.
We start from the observation that, in the response described above, the area denoted by Ψ in Fig. 1a represents recoverable work, since this area vanishes when unloading proceeds up to the origin. Moreover, the area denoted by ∆ represents an irreversibly lost work, since no point in this area is accessible from the point (u(t), α(t)). Accordingly, for any state (u,α) we take the area Ψ to define the elastic strain energy
Ψ(u,α) = 21 g(α) u2 , (6) and we take the area ∆ to define the dissipated work
∆(α) =
α 0
) (s ds
f − 12g(α) α 2 . (7) This is the work spent in a loading-unloading cyclic process from u = 0 to u = α and back. It can be deduced from the dissipation potential
Φd(α,α) = −21 g'(α)α 2α . (8) Indeed, the dissipation power associated with Φd
Dd(α, )α = α α∂∂ Φd(α,α) = −21g'(α) α 2 α , (9) coincides with the time derivative ∆(α)of ∆(α). We note that this coincidence is a characteristics of rate independent materials. Indeed, the response defined above is insensitive to time rescaling. Note also that the derivative of Φd with respect to α is the generalized force associated with the generalized velocity α, see e.g.
(Halphen and Nguyen 1975).
In any process t u(t), the work performed in a time interval (t1, t2) is equal to the sum of the changes in the energy and the dissipation over the same interval:
) ( ) ( ) ( ) ( ) ,
(t1t2 t2 t1 t2 t1
W =Ψ −Ψ +∆ −∆ . (10) This equation expresses the conservation of energy, in the absence of applied loads and when inertia effects are negligible. This equation and the dissipation inequality
Dd(α, )α ≥ 0 , (11)
which is the purely mechanical version of the Clausius-Duhem inequality, see e.g. (Truesdell & Noll 1965, Sect.
79), are the mechanical counterparts of the first two laws of thermodynamics. The inequality is satisfied if and only if
α ≥ 0 . (12) Indeed, if α > 0, the inequalityα ≥ 0 follows from the negativeness of g'(α). If α = 0, the same inequality follows from the fact that α cannot decrease, because it cannot take negative values.
By time differentiation of (10) we obtain the power equation ) ( ) ( )
(t t t
P =Ψ +∆ , (13) which is therefore a necessary condition for energy conservation. In the present model, the external power is
P(t) = σ(t) u(t) , (14) the time derivative of Ψ is
Ψ(u,α) = 21g'(α) u2 α + g(α) u u, (15) and the time derivative of ∆ is given by (9). With Ψ, ∆, P as in (6), (7), (14), the power equation becomes
α α α α
σ ( ) ) (' )( )
( −g u u+12g 2−u2 = 0 . (16) Due to the restrictions on the sign of each factor, the term g('α)(α2−u2)α is non-positive. Then, one has
u u g( ) )
(σ− α ≥ 0 (17) for any u. In particular, if u > 0, the sign of u is free, and one has
σ = g(α) u ≥ 0 if u > 0. (18) In this way, the relation (3) between σ, α, and u when 0 < u < urhas been re-obtained. At u = 0, one is free of choosing any constitutive relation compatible with inequality (17). In view of the requirement (v) about the unilateral contact, we introduce the decomposition σ = σ +− σ −, with
σ + = max {σ , 0 } , σ − = max {−σ, 0 } , and for σ + we take the constitutive equation
u g(α)
σ+ = . (19) For σ − we observe that, when σ < 0 and u = 0, inequality (17) reduces to −σ−u≥ 0. Because u≥ 0 by (1) and σ− ≥ 0 by its very definition, it must be σ−u= 0. In this way, the Signorini unilateral contact law
0 ,
0 ,
0 ≥ =
≥ −
− u σ u
σ , (20) is obtained. In particular, this law tells us that σ −= 0 for all u > 0. When u = 0 and u> 0, it follows from (17) that σ ≥ 0, and therefore σ −= 0. This leads to the unilateral contact law for the velocities
0 ,
0 ,
0 ≥ =
≥ −
− u σ u
σ if u = 0 . (21) The two laws together tell us that σ − is non-zero only if u =u= 0.
From equation (19) we have that σ−g(α)u=0when σ ≥ 0, and from (21) we have that u= 0 when σ < 0.
Hence, the equality(σ −g(α)u)u= 0 holds in all cases. The power equation (16) then reduces to
g′(α)(α2−u2)α = 0 . (22) To determine the evolution law for α, we partition the state space into the four subsets listed below, and we determine separately the evolution of α in each subset.
(i) In the subset {0 ≤ u < α < ur}, both g′(α) and (α2 −u2) are different from zero. Therefore, equation (22) gives α = 0. The interface evolves according to the linear elastic law σ = g(α) u, with constant stiffness g(α).
(ii) In the subset {0 < u = α < ur}, equation (22) is satisfied identically. A necessary condition for energy conservation is then that the second time derivative of equation (10), that is, the first time derivative of (16), be zero:
((σ −g(α)u)u)·+12(g'(α)α)· (α2−u2)+ g'(α)(αα−uu)α = 0 , (23) and sinceσ =g(α)uand g′(α) < 0 at both the current and the following instants, this equation reduces to
0 )
(α−u α = . (24) For u≤ 0, condition α ≥ 0 implies α = 0. If u> 0, from the observation that α = u at the current instant and α ≥ u at all instants implies α ≥ u, we deduce that α = u.
(iii) At the point {0 = u = α } equations (22) and (24) are satisfied identically, and the vanishing of the next time derivative of the energy is required . By differentiating equation (23) at α = u = 0, one obtains
0 )
(α2−u2 α = , (25) and because α = u = 0 implies α ≥ u, it follows that α = u.
(iv) Finally, in the subset {α = ur} α is zero, because α cannot decrease because of the dissipation inequality α ≥ 0, and cannot increase because it cannot take values greater than ur.
The above conclusions can be collected in the formula
α = 0u ifα =uotherwise<ur and, u>0, (26) which specifies the evolution of α in all possible cases. This is an evolution law of the same type as the Drucker- Prager law in plasticity. We have shown that this evolution law is determined by the assumptions u ≥ 0 and u ≤ α ≤ ur, the general laws (10), (11), the choices (6) for the elastic potential and (8) for the dissipation potential, and the constitutive assumption (19).
2.2. Adhesion with purely normal damage and viscosity
In the presence of viscous dissipation, the points of the (σ, u) plane located above the loading curve become admissible. Then the restriction u ≤ α is removed, and the state space becomes
Σ = { (u, α) | u ≥ 0 , 0 ≤ α ≤ ur } . (27) A consequence of this enlargement of the state space is that the state variable α is no longer identified with the largest displacement u reached before the current instant.
For the strain energy we keep the expression (6), and for the dissipation potential due to damage we keep the expression (8). To it, we add the potential of the viscous dissipation
Φv(α, )α = 41h(α)α2, (28) with h(α) > 0. With this potential is associated the dissipation power
Dv(
α
,α) = ∂∂αΦv(α, )α α = 21h(α) α2. (29) The total dissipation power is then
D = Dd + Dv , (30) with Dd as in (9). Notice that both contributions are governed by the same state variable.
The dissipation inequality Dd ≥ 0 still yields α≥ 0, and the power equation (16) now has an extra term due to viscous dissipation:
α α
α α
σ ( ) ) '( )( )
( −g u u+21g 2−u2 −21 h(α)α2 = 0 . (31) We keep the constitutive equation (19), which leads to the unilateral contact conditions (20). Then the power equation reduces to
0 ) ( ) )(
(
'α α2−u2 α−hα α2=
g . (32) Let us exclude the trivial solution of αidentically zero, which corresponds to a non-dissipative response. Then we determine separately the evolution of α in the three following subsets of the state space:
(i) In the subset{α ≤ u, 0 ≤ α < ur}, the differential equation ) ) (
(( )
' 2 u2
h
g −
= αα α
α (33) holds. Indeed, the condition α≥ 0 is satisfied because g′(α) / h(α) < 0 and α 2 ≤ u2. Moreover, there is dissipation,α> 0, only if α 2 < u2.
(ii) In the subset{u < α, 0 ≤ α < ur}, one has h(α)α2=g'(α)(α2−u2)α≤ 0, with h(α) > 0. Then it must beα = 0.
(iii) In the subset{α = ur} one hasα= 0, because α cannot decrease by the dissipation inequality α ≥ 0, and cannot increase because it cannot take values greater than ur.
In conclusion, the evolution law is
α = − − < ≤ <
. otherwise 0
, 0
, if ) )(
( )
(' 2 2
ur
u h u
g αα α α α
(34) This law has been obtained with the same assumptions made in the non-viscous case, except that the assumption u ≤ α has been removed, a viscous dissipation potential has been added, and the trivial case of α identically zero has been excluded.
The dotted line in Fig.2a is the response curve for the cyclic process shown in Fig. 2c, starting from the initial state (u(0),α(0))= (0,α0). In any response curve, the slope g(α) of the line joining the origin to the point (σ, u) cannot increase in time. Indeed, its time derivative g′(α)α is non-positive. Therefore, just like in the non- viscous case, the points located above the line σ = g(α) u are not accessible from the point (σ, u).
Figure 2. Adhesion with damage and viscosity. Response curve in the (σ , u) plane, (a), and in the state space, (b), for the cyclic process shown in (c), under the initial condition α(0) = α0 .
Consider a point (σ, u) located above the loading curve σ = f(u). For any such point, u is greater than α. Then for any response curve crossing this point, by differentiation of (18) one gets
u g h u
u u g
g u g u
g ( ) ( )
) (
) ( ) ' ( )
( ' )
(α α α α 2αα 2 α2 α
σ = + = − − ≤ . (35) This inequality shows that the slopeσ/uof the curve is never greater than the current elastic modulus g(α) at loading (u> 0), and is always greater than the same modulus at unloading (u≤ 0). Consequently, every response curve located above the curve σ = f(u) is star-shaped with respect to the origin. If, as in the case shown in the figure, there are several such curves, their union is star-shaped with respect to the origin as well.
The figure also shows the dissipation associated with the given process: the dissipation due to damage is given by the area ∆d, and the dissipation due to viscosity is given by the area ∆v. While ∆d is a function of the current state as in the non-viscous case, ∆v depends on the velocityu.
The state space Σ is the disjoint union of the two regions
Σe = {(u,α)∈Σ | u < α, 0 ≤ α ≤ ur} , Σd = {(u,α)∈Σ | α ≤ u, 0 ≤ α ≤ ur} . (36) As in the non-viscous case, Σe is the region of the non-dissipative processes, α = 0. The dissipative processes, (α > 0), which in the non-viscous case were confined to the line u =α , now spread over the whole region Σd. The dotted curves in Fig. 2b are the counterparts of the response curves in Fig. 2a. By the evolution law (34), in the region Σd the slope dα/du = α/ u increases with the distance (u2 −α2) from the region Σe, and is inversely proportional to the loading rate. In the limit case u→ +∞ the slope is zero; this means that there is no dissipation in infinitely fast processes. In the other limit case u→ 0 the slope is +∞. The corresponding slopes in the (σ, u) plane can be deduced from equation (35). They are: σ/u= g(α) when u→ +∞, and σ/u= −∞
when u→ 0. The latter case describes the stress relaxation which occurs at fixed deformation.
(a) (c) (b) σ = f (u)
α0 α1 α2 α3 um ur
u σ
∆∆∆∆v
∆∆∆∆d
O
α3 α2
α1
α0
α
um ur u ΣΣΣΣd
ur
ΣΣΣΣe
t um
u
0
2.3. Adhesion with normal and tangential damage
In the presence of tangential forces τ and of tangential displacements v we keep a single state variable α, and we define a state of the interface to be the triplet (u, v, α). For u we keep the non-interpenetration condition u ≥ 0, and for α we keep the restrictions 0 ≤ α ≤ αr, where αr is the critical value which determines the complete rupture of the interface.
The normal response σ = fN(u) and the tangential response τ = fT(ν) of the interface are described by two constitutive functions, fN and fT . We assume that fN is defined over the non-negative u, and that it is positive in (0,αr) and null outside. The function fT is defined over the whole real line and is odd, fT(−ν) = − fT(ν), positive in (0,αr), and null outside (−αr,αr). We also assume that the restrictions of fN and fT to (0,αr) are star-shaped with respect to the origin, so that the functions
gN(α) := fNα(α) , gT(α) := fTα(α) , (37) are strictly decreasing when 0 ≤ α < αr.
The strain energy is taken to be the sum of a normal and a tangential part
Ψ(u, v, α) = 21gN(α) u2 + 21gT(α)ν 2 , (38) and the same holds for the dissipation potential
Φd(α, )α = −21(g'N(α) + g′T(α))α 2α . (39) The latter corresponds to the dissipated work
∆(α) = +
α 0
)) ( ) (
(fN s fT s ds −21(gN(α) + gT(α))α 2 . (40) and to the dissipation power
Dd(α) = ∆(α) = −12(g′N(α) + g′T(α)) α 2α
.
(41) Note that since both gN and gT are strictly decreasing, the dissipation inequality Dd(α)≥ 0 still reduces toα ≥ 0 when α > 0. The power equation now becomesσu+ τν = gN(α)uu+ gT(α)vv+21 gN'(α) (u2−α2)α +21 gT'(α) (v2−α2)α. (42) We take the constitutive laws
u gN(α)
σ+= , τ = gT(α) v , (43) keeping the possibility of a negative normal force σ − when u = u= 0, see equations (20), (21). With these laws, equation (42) reduces to
(
gN'(α) (u2−α2) + gT'(α) (v2−α2))
α = 0 , (44) and after settingρ(α) := '( ) '( ) ) (
' α
α α
T N
N g g
g
+
,
ϕ(u, v, α) := ρ(α) u2 + (1 −ρ(α)) v2 − α2 , (45) the power equation takes the formϕ(u, v, α) α = 0 . (46) The state space is now
Σ = { (u, v, α) | u ≥ 0 , 0 ≤ α ≤ αr , ϕ(u, v, α) ≤ 0 } . (47) This is a three-dimensional region, whose intersections with the planes α = const are half-ellipses. At the interior points of Σ the strict inequality ϕ(u,v,α) < 0 holds, and equation (46) yieldsα = 0. Therefore, there is no dissipation at the interior of Σ.
On the portion of the boundary at which ϕ(u,v,α) = 0, equation (46) is satisfied identically. To determine αwe follow the same procedure as in the purely normal case. From the vanishing of the second derivative of the energy, we obtain
(
ϕu(u, v, α)u+ ϕv(u, v, α)v+ ϕα(u, v, α)α)
α = 0 , (48) where the subscripts denote partial derivatives. We make the supplementary assumptionϕα(u, v, α) < 0 ∀α > 0 , (49) under which the term between parentheses in (48) is strictly negative when ϕu(u, v, α)u+ϕv(u, v, α)v≤ 0. In this case the equality is satisfied only if α = 0. If ϕu(u,v,α)u+ϕv(u,v,α)v is positive, α is determined by equating to
zero the term between parentheses. Therefore, at the portion of the boundary at which ϕ(u,v,α) = 0 we have the evolution law
α = + + >
−
otherwise.
0
, 0 ) , , ( ) , , ( ) if
, , (
) , , ( ) , ,
( u v u u v v
v u
v v u u v
u v u v
u ϕα ϕα α ϕ α ϕ α
ϕ
α (50)
This law has the following geometrical interpretation: the vector (ϕu,ϕv,ϕα) is the exterior normal to the boundary, and its projectionN =(ϕu,ϕv,0)on the plane α = const is the exterior normal to the intersection between the same plane and Σ. The assumption ϕα< 0 implies that the elliptic sections of Σ expand with increasing α, in such a way that the section at any α0 is strictly contained in the section at α for all α > α0. In the language of plasticity, we are assuming an isotropic hardening law.
Let U =(u,v)be a given increment of U = (u,v). According to the law (50), if U points inwards Σ, U
v u
N( , ,α)⋅ ≤ 0, the evolution occurs at constant α, that is, without dissipation. If U points outwards, U
v u
N( , ,α)⋅ > 0, the response is dissipative, with
α =− ( , , ) ) , ,
( α
ϕ α
α uv U v u
N ⋅ . (51) Equation (48) tells us that in the dissipative case the vector (u,v,α)is orthogonal to the gradient of ϕ, and, therefore, tangent to the boundary. Thus, the dissipative response takes place on the boundary of Σ.
For α = 0, from (45)2 and (47) it follows that u = v = 0, so that the cross-sectional ellipse reduces to a point.
But at this point the gradient of ϕ is zero, and equation (48) is satisfied identically. Then it becomes necessary to consider the third derivative of the energy equation (10), that is, the derivative of (48). It involves the second gradient of ϕ , which at u = v = α = 0 is
ϕuu = 2 ρ(0) , ϕvv = 2 (1 − ρ(0)) , ϕαa = −2 , ϕuv = ϕuα= ϕvα= 0 . (52) The corresponding evolution law is
2=
α ρ(0) u2 + (1−ρ(0)) v2. (53) For α = αr, we have α=0 as in the purely normal case. Therefore, the overall evolution law for the state variable α is
α = + − =
>
⋅
=
<
⋅ <
−
otherwise.
0
, 0 if
(0)) (1 (0)
, 0 ) , , ( , 0 ) , , ( , 0
) if , , (
) , , (
2
2 ρ α
ρ
α α
ϕ α α α
ϕ α
α
v u
U v u N v
v u u
U v u
N r
(54)
Let us briefly discuss the physical meaning of the constitutive functions fN and fT, and how to determine them experimentally. In the purely normal case, the equation of the loading curve was σ = f(u), and f was determined by a monotonic loading test. In the present case of coupled normal and tangential actions, one may consider a purely normal loading process with increasing u and null v, as well as a purely tangential loading process with increasing v and null u. Each process determines an evolution curve on the boundary region ϕ(u,v,α) = 0 of the state space Σ. For the first process, equation (45)2 yields
0 = ϕ(u, 0, α) = ρ(α) u2 − α2 , (55) which means that u = αρ −1/2(α) instead of u = α as in the purely normal case. Therefore, the response curve σ = gN(α)u does not coincide with the constitutive curve σ = gN(u) u = fN(u). Similarly, the curve τ = gT(α)v does not coincide with the constitutive curve τ = gT(v) v = fT(v). However, taking an increasing “diagonal” loading with u = v, equation (45)2 reduces to the equality
0 = ϕ(u, u, α) = u2 − α2 , (56) from which it follows that u = v = α, and therefore σ = gN(u) u = fN(u) and τ = gT(v) v = fT(v). Therefore, the constitutive functions fN and fT are determined by this special loading process. This means that, as anticipated in the Introduction, in the presence of both normal and tangential deformation the constitutive functions are not determined by two separate loading experiments, one for the normal and one for the tangential deformation, but by a single diagonal experiment.
2.4. An example of evolution under complex loading
Consider an interface subject to normal and tangential damage, in the initial state u = v = α = 0. As an example, we determine the response of the interface to the process t (u(t), v(t)) made of the following five steps:
, to from ,
0 to 0 from }
5 {
, 0 to 0 from ,
0 to from }
4 {
, 0 to from ,
to from }
3 {
, 0 to 0 from ,
} 2 {
, 0 ,
0 to 0 from }
1 {
4 5 4 ) ( 5
) (
4 )
( 4
3 ) (
3 2 ) ( 2 3 2 ) (
2 )
( 1
2 ) (
1 ) ( 1
) (
v v v v
u u
v v
u u u
v v v
u u u u
v v
u u u
v v u
u
t t
t t
t t
t t
t t
<
=
<
=
=
>
>
=
=
=
=
>
(57)
each of which performed at constant speed. For the sake of simplicity we assume that αr = +∞, and we consider the special case of gT and gN proportional, gT(α) =λ gN(α), in which ρ(α) takes the constant value ρ = (1+λ)−1. In this case, the region Σ is a cone with vertex at the origin. The cone is shown in Fig. 3a, where the process (57) is represented by the dotted line on the plane (u,v). For constant ρ, the gradient of ϕ reduces to
ϕu = 2ρ u2, ϕv = 2 (1−ρ)v2, ϕα= − 2α , (58) and the evolution law (50) for the boundary points takes the simplified form
α
α = ρuu+ (1−ρ)v if v ρuu+ (1−ρ)vv> 0 , α =0 otherwise. (59) If ρu(t)u+ (1−ρ)v(t)v> 0 over the time interval (t0, t), by time integration we get
α(t)=
(
α2(t0) + ρ (u2(t)−u2(t0))+ (1−ρ) (v2(t)−v2(t0)))
1/2. (60) The first step of the given process starts at the vertex of the cone with initial velocities u>0 and v=0. From equation (53) it follows that α = ρ (0) 1/2u(0), and a loading regime starts. At the subsequent instants, from (60) one has α(t)= ρ 1/2u(t), and at the end of the step one getsα1 = ρ 1/2u1 . (61) In the second step, since u=0 and vv> 0, a new loading regime takes place. Again from (60) it turns out that, at the end of the step,
α2 =
(
ρ u22+ (1−ρ)v22
)
1/2 . (62) The third step has u>0 and vv< 0. The assumption that the speed is constant throughout the step determines the proportionality relations2 3
2 2
) (
2 ) (
u u v u
u v v uv
t
t −− = −−
= . (63)
Figure 3. Evolution of α during the process (53) (a), and (b) the three possible patterns of evolution for the third step (in (a), the scale of α has been amplified).
α3=α4
α2
α1
3 4
1
α
v u
α5
5
2 0
vΣ
(a) (b)
P
u
u2
H
K L
v2 v α2 ρ−1/2
α2 (1−ρ)−1/2
Then in equation (48) the term
ϕuu+ ϕvv = ρu(t)u+ (1−ρ)v(t)v (64) has the same sign as the difference
ρ u(t) (u3 − u2) − (1−ρ) v(t) v2 . (65) Therefore, it is the sign of this term which determines whether there is loading or unloading. In particular, at the initial instant of the third step there is loading if
ρ u2 (u3 − u2) − (1−ρ) v22≥ 0 , (66) and because the difference (65) increases with t, it remains positive during the whole step. This is the case represented by the line PH in Fig. 3b. On the contrary, if
ρ u32≤ α22 =ρ u22+ (1−ρ) v22 , (67) then ϕ(u3,0,α2) =ρ u32 −α22 < 0, that is, the segment joining the points (u2,0,α2), (u3,0,α2) belongs is interior to the state space Σ. In this case, the unloading regime represented by the line PL in the figure takes place. In the intermediate case
ρ u2 (u3 − u2) < (1−ρ) v22 < ρ (u32− u22) , (68) there is initial unloading, followed by a loading regime. The transition from unloading to loading occurs when the line PK intersects the ellipse. We consider the case (66) of loading over the whole step. At the end of the step, one has
α3 = ρ1/2u3 . (69) In the fourth step, one has u<0 and vv> 0. Since the initial value of v is zero, the term involving uis initially dominant. Therefore, an unloading regime takes place. If |v4| < α3 (1−ρ)−1/2, this regime is maintained over the step, and the final value of α is α4 =α3.
In the fifth and last step, one has u=0 and vv> 0. Since the step starts at the interior of Σ, unloading will occur until the point (0,v(t)) reaches the boundary of Σ. This occurs when ν = vΣ=−α4 (1−ρ)−1/2. If | v5 | > | vΣ |, a loading regime follows, at the end of which one has
α5 = − (1−ρ) v5 . (70)
Figure 4. Normal and tangential response curves in the case of an interface subjected to the process (53) (the gray areas are the energies dissipated in the first three steps).
We now determine the response curves (σ, u) and (τ, v). In the first step, since it was established that α(t)= ρ1/2u(t), it follows from the constitutive equations (43) that
σ = u gN(ρ 1/2u) , τ = 0 . (71) The first equation provides the expression for the loading curve for u when v = 0. This curve can be drawn up as shown in Fig. 4a. For a given u1 and for α1=ρ 1/2u1, the intersection P1 between the constitutive curve σ = fN(u) and the vertical from α1 determines the slope gN(α1), and the intersection H1 between the line OP1 with the vertical from u1 determines a point on the loading curve. The entire loading curve can be drawn up by repeating this procedure for a sufficiently large number of values of u.
τ
0=v1=v3 α1 α2 v2 α3 v K2
τ = fT(v) τ = v gT((1−ρ)1/2v)
∆∆∆∆T2
∆∆∆∆T3
∆∆∆∆T5
v5 -α5 vΣ -α4 v4
K4
K5
O=K1 =K3
σ = u gN(ρ1/2u) σ = fN(u)
∆∆∆∆N1
σ
α1 α2 u1=u2 α3 u3 u
∆∆∆∆N2
∆∆∆∆N3
P1
H1
H2
H3
O=H4 =H5
0=u4=u5
In the first step, the loading curve (71)1 is followed up to the point H1, while in the (τ, v) plane there is no evolution because τ(t) = v(t)= 0. In the second step, the response curve τ= gT(α)v starts at the origin with the slope gT(α1), and ends at the intersection K2 between the vertical from v2 and the line from the origin with slope gT(α2), with α2 given by (57). In addition, since u is constant, the response curve σ = gN(α) u is the vertical segment starting from H1 and ending at the intersection H2 with the line from the origin with slope gN(α2).
In the third step, assuming that inequality (66) holds, there is a loading regime, in which the response curve for τ starts at K2 and ends at the origin, while the response curve for σ starts at H2 and ends at the intersection H3 between the vertical from u3 and the line from the origin with slope gT(α3), where α3= ρ1/2 u3is given by equation (69).
In the fourth step, there is an unloading regime. The two response curves lie on straight lines from the origin.
More precisely, the curve σ goes through the segment H3O, and the curve τ starts from the origin with the slope gT(α3), which is the slope of the tangent to the curve at the end of the previous step. In the last step, the point (σ, u) stays fixed at the origin, and the point (τ, v) goes through the same line as before up to v = vΣ, and then continues along the loading curve τ= v gT((1−ρ)1/2v) up to the point K5.
In Fig. 4 it can be seen that the state variable α is given by the abscissas of the intersections of the curves σ = fN(u) and τ = fT(u) with the lines connecting the origin to the points (σ , u) and (τ , v), respectively. The areas below the response curves give the work performed. The dissipated energy is given by the gray areas: ∆N1 in the first step, ∆N2 + ∆T2 in the second, ∆N3 + ∆T3 in the third, and ∆T5 in the fifth step.
2.5. Adhesion with damage, viscosity, and friction
We now generalize the model with normal and tangential loading by introducing viscosity and friction. A state of the interface is still defined by the triplet (u, v, α). Viscosity is accounted for by adding the viscous dissipation power Dv given by (29). For friction, we add the frictional dissipation power
Df (α, σ −, )v =µ(α) σ − ||v , (72) where σ − is the contact pressure exerted on the interface, and µ is a function which measures the intensity of friction. We assume that µ is an increasing function, with µ(0) = 0 and µ(αr) = µ∞. Accordingly, friction gradually comes into the play when the adhesion decreases, and is fully operating only after the adhesion disappears. We recall that, because of conditions (16) and (18), the pressure σ − may take arbitrary positive values if u = u= 0, and is zero otherwise. In the present model, when u = u= 0 the pressure σ − is assumed to be a known applied load. Keeping the expression (38) for the strain energy and (41) for the dissipation power due to damage, the power equation becomes
σ u+τν = gN(α)uu+ gT(α)vv+12gN'(α) (u2−α2)α +21gT'(α) (v2−α2)α + 12h(α)α +2 µ(α)σ −|v|. (73) For σ we keep the constitutive law σ + = gN(α) u, while τ is assumed to be the sum of two contributions, one due to damage and one due to friction:
τ = τd + τf , τd = gT(α)v , τf =µ(α) σ − sgnv. (74) With the constitutive assumptions just made, equation (73) reduces to
(
gN'(α)(u2−α2) + gT'(α)(v2−α2))
α + h(α)α = 0 2 . (75) In addition, taking ρ as in (45) and settingκ(α) := − '(α)(α)'(α)
T
N g
g
h+
,
(76) equation (73) further reduces to(
ρ(α) u2 + (1−ρ(α)) v2 − α2)
α −κ(α)α = 0 2 . (77) It follows that α = 0 if the term between parentheses is non-positive, and that) (1
α
α =κ
(
ρ(α) u2 + (1−ρ(α)) v2 − α2)
(78) if the same term is positive. The state spaceΣ = { (u, v, α) | u ≥ 0 , 0 ≤ α ≤ αr } (79) is the disjoint union of the regions
Σe = { (u, v, α)∈Σ | ρ(α) u2 + (1−ρ(α)) v2 ≤ α2, α ≤ αr } ,
Σd = { (u, v, α)∈Σ | ρ(α) u2 + (1−ρ(α)) v2 > α2, α ≤ αr } , (80) Σr = { (u, v, α)∈Σ | α =αr } .
In the first region we haveα = 0, while α is positive and given by (78) in the second. Therefore, Σeis the region in which the non-dissipative processes occur, and Σd is the region where the dissipative processes occur. In the region Σr we have gN(α) = gT(α) = 0 and, therefore, σ = τ = 0. This is the region of total rupture, at which α = 0.
Moreover, when the conditions u = u= 0 for unilateral contact are satisfied, the evolution of α is governed by equation (78) with u = 0, and the response τ is determined by equations (74), with σ − a given function of time. Thus, the evolution law for α is fully determined.
As an example, let us determine the evolution of α in the process (57) and from the initial state (0, 0, 0), in the presence of viscosity and friction. The functions ρ, κ and µ are assumed to be constants, and αr is taken to be equal to +∞. As shown in Subsection 2.4, ρ = const implies that gN and gT are proportional. Moreover, by (76), κ
= const implies that the viscosity h is proportional to g′N + g′T.
Since the response is now rate-dependent, the initial time t0 = 0 and the final instants t1 ... t5 of all loading steps must be specified. If at each step the displacement rates ui, are constant, one has vi
u(t) = ui−1 + (t − ti−1) ui, v(t) = vi−1 + (t − ti−1) vi, ∀t ∈( ti−1, ti) , (81) and the evolution law (72) gives
κα(t)+α2(t) = ρui2−1+(1−ρ)vi2−1+ 2(ρui−1ui+(1−ρ)vi−1vi)(t − ti−1) + (ρui2+(1−ρ)vi2)(t − ti−1)2 . (82) By integrating this differential equation under the initial condition
α(ti−1) = αi−1 , (83) the evolution of α during the ith time interval can be determined. For example, in the first step one has t0 = u0 = v0
= α0 = 0 andv1= 0. Then equation (82) reduces to the Riccati equation
κα(t)+ α2(t) = ρui2t2 , (84) subject to the initial condition α(0) = 0. The solution of this problem determines the initial condition α1 for the next step. One may keep proceeding in this way as long as the trajectory t (u(t), v(t), α(t)) lies inside the region Σd. This is the case in our example during the first three steps. In the fourth step, the direction of the process t (u(t), v(t)) is such that the trajectory enters Σe at a time tΣ. The process then continues with constant α up to the state (0, v4, α4), with α4 = α(tΣ).
Figure 5. Evolution of α during the process (57), in the presence of viscosity and friction (the scale of α has been amplified).
1 2
α3
α2
α1
α
vΞ 3
4
v u
α5
5
0
α4
vΣ
